I see many mathematicians conflating the definitions of traveling waves and solitons, and I am unable to understand, from a mathematical point of view, the differences between these two types of solutions for a nonlinear dispersive PDE. All I know is the following:

Consider for example a nonlinear dispersive PDE which is [completely integrable][1], i.e. has infinite conservation laws, ( I think the property of complete integrability will not probably add something to the question)

- The **traveling waves** are solutions of the form $u_0(x+ct)$ where $u_0$ is the initial data and $c\in\mathbb{R}$. 

- The **Solitons** are subset of the traveling waves, that remain with the same shape even after colliding with another soliton. The phenomena of solitons appear after a cancelation between the dispersive effects and the nonlinearity of the equation.

So here are my questions:

 1. How do we know if a nonlinear PDE has solitons as solutions, knowing that it has traveling wave solutions ? On other words, how do we prove mathematically that a traveling wave is a soliton (without using simulation).
2. For example, solutions of the form $u(t,x)=e^{i(x-t)}$ or $u(t,x)=\frac{1}{1-\frac12 e^{i(x-ct)}}$, $x\in \mathbb{T}:=\mathbb{R}/(2\pi\mathbb{Z}),$ can be considered as solitons?
3. Does a traveling wave that is almost periodic solution, *i.e. the set $\{u(\cdot+\tau), \tau\in \mathbb{R}\}$ is relatively compact*, can lead to the fact that it is a soliton ?



  [1]: https://en.wikipedia.org/wiki/Integrable_system